It's hard to give an authoritative answer because it depends on how your bank carried out the changes to your old and new cards. And, on how you're using the card(s).
Ultimately, credit cards are just a tool for accessing a credit card loan. You can cancel, replace, lock, etc the card without closing or otherwise impacting the loan. You can even change the terms of the loan (change the interest rate, change the limit, etc) independently from "closing" the loan. This is important because it's the loan That shows up on your credit report, not literally the card itself.
It sounds like you've effectively replaced the old card with a new one. If the bank closed the old loan and opened a new one, you'll see that on your credit report - the old account will show up as closed and the new one will appear as new. This will impact your credit score in a handful of ways in terms of credit mix, average age of credit, utilization, and other factors.
However, instead, they may have just kept your old loan open, changed the terms, and issued a new card. If this was the case, there may still be an impact, but it may be much smaller. For instance, if they changed your credit limit, and you're carrying a balance, your utilization rate will change. But if all they changed were the annual fee and the interest rate, that won't impact your score at all.
In a comment, you clarified that the other card was one that already existed, and they're "transferring the limit" from the closed card to the new one. This will have a slightly different set of implications:
- Closing the old card will me that your old credit card loan account will now show up as closed. As it ages off your report, your average age of credit will change (whether this has a net positive or negative impact will depend on how old this account was compared to your other accounts).
- Similarly, the closed account will age out of the credit mix factor. Depending on what other types of credit accounts you have open right now, this may not make much of a difference at all, especially since you have another credit card loan (so you're not "losing" a credit type).
- The credit limit and balance on this closed account will no longer count in utilization rate. Again, this may be a net positive or negative impact, depending on your other credit utilization. If you never carry any balance on any cards, the impact will be zero. If you had a low balance and high limit on this card, but high balances on other cards, your score will drop. If your utilization on this card was worse than your others, your score will increase.
- Them "transferring" the limit implies that they're raising the limit on your other card. This will also affect utilization, but again the magnitude and direction of the impact will depend on your overall utilization of this and your other accounts (and, again, if you never carry balances, your impact will be zero).
And, finally, it's important to note that the soft pull they did won't impact your score.